Optical Properties of Solids LM Herz Trinity Term 2014

Transcription

1 Optical Properties of Solids LM Herz Trinity Term 2014 Contents: I. Absorption and Reflection II. Interband optical transitions III. Excitons IV. Low-dimensional systems V. Optical response of an electron gas VI. Optical studies of phonons VII. Optics of anisotropic media VIII. Non-linear optics Recommended textbooks M Fox, Optical Properties of Solids, Oxford University Press PY Yu and M Cardona, Fundamentals of Semiconductors, Springer C Kittel, Introduction to Solid State Physics, Wiley B Saleh, M. Teich, Fundamentals of Photonics, Wiley A Yariv, Quantum Electronics, Wiley TT2014 1

2 Interaction of Electromagnetic Radiation with Matter reflected light Absorption MEDIUM transmitted light Scattered light (phonons, impurities...) incident light Luminescence TT2014 2

3 I Absorption and Reflection Macroscopic Electromagnetism Maxwell s equations: (no net free charges) (non-magnetic) (ohmic conduction) Linear Optics In a linear, non-conducting medium: Solution: where: complex! Define complex refractive index: refraction absorption TT2014 3

4 Absorption Intensity decay of wave: (Beer s law) Define absorption coefficient: At normal incidence: Reflection Reflectivity R for a material s surface contains information on its absorption! Relationship between components of ñ and ε r and if (weak absorption): TT2014 4

5 The classical dipole oscillator model Inside a material the electric field of an EM wave may interact with: bound electrons (e.g. interband transitions) ions (lattice interactions) free electrons (plasma oscillations) Equation of motion for a bound electron in 1D: where stationary solutions: Displacement of charge causes polarisation: (N: oscillator density) background TT2014 5

6 Real and imaginary part of ε r Can now calculate and Optical constants for a classical dipole oscillator TT2014 6

7 Local field corrections In a dense medium: atoms experience local field composed of external field E and polarization from surrounding dipoles treat interacting dipole as being at centre of sphere surrounded by a polarized dielectric Clausius-Mossotti relationship: electric susceptibility per atom Problems with the classical oscillator model no information on selection rules need quantum mechanics interband transitions should depend on the density of states g(e) One possible modification: oscillator strength (from QM) write: line shape of transition (from classical oscillator model) TT2014 7

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9 II Interband optical transitions conduction band f i valence band Treat interband transitions through timedependent perturbation theory - Fermi s golden rule gives transition probability: joint density of states with matrix element: where dipole moment E-field of incident wave Bloch wavefunctions Matrix element for interband transitions: absorption deduce conditions for dipole allowed (direct) transitions. emission Consider: (1) wavevector conservation (2) Parity selection rule (3) dependence on photon energy TT2014 9

10 Conditions for direct interband transitions (1) wavevector conservation only if or absorption emission E typically: vertical transitions E g k (2) parity selection rule odd parity only if and have different parity! In a typical 4-valent system (e.g. group IV or III-V compound): p isolated atom sp3-hybrid p-antibonding s-antibonding in crystal conduction band s p-bonding E g valence band s-bonding expect to see strong absorption for these materials TT

11 (3) Dependence of transition probability on photon energy Final state is an electron-hole pair where reduced effective mass If M if is independent of the photon energy ω, the joint density of states contains the dependence of the transition probability on ω. For this case: Absorption coefficient (for direct transitions): Examples for direct semiconductors a) InSb at 5K b) GaAs at 300K data deviations from e.g. due to phonon absorption or non-parabolicity of the bands. TT

12 Indirect interband transitions Indirect gap: valence band maximum and conduction band minimum lie at different wavevectors, direct transitions across the indirect gap forbidden, but phonon-assisted transitions may be possible. (i) wavevector conservation: E with phonon wavevector q phono n E g k (ii) probability for indirect transitions: perturbation causing indirect transitions is second order optical absorption much weaker than for direct transitions! find absorption coefficient phonon absorption or emission TT

13 Example for an indirect semiconductor: Si Band structure: Absorption spectrum: indirect transitions mostly direct transitions E 1 E g E 2 E 2 E 1 E g TT

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15 III Excitons Absorption coefficient of CuO 2 at 77K: Series of absorption peaks just below the energy gap Coulomb interaction between electron and hole gives rise to excitonic states (bound electron-hole pairs) Wannier-Mott Excitons Frenkel Excitons e e h h weakly bound (free) excitons binding energy ~ 10meV common in inorganic semiconductors (e.g. GaAs, CdS, CuO 2...) particle moving in a medium of effective dielectric constant ε r strongly (tightly) bound excitons binding energy ~ 0.1 1eV typically found in insulators and molecular crystals (e.g. rare gas crystals, alkali halides, aromatic molecular crystals) particle often localized on just one atomic/molecular site TT

16 Weakly bound (Wannier) Excitons Separate exciton motion into centre-of-mass and relative motion: CM motion: exciton momentum: where exciton mass: kinetic energy: Relative motion: Binding energy: where (Rydberg) reduced mass Exciton radius: where a 0 = 0.529Å (Bohr radius) E-k diagram for the weakly bound exciton (a) uncorrelated electron-hole pair (one-electron picture) E (b) exciton (one-particle picture) E wavevector conservation: ; k wavevector conservation:... R X n=2 E n=1 g k transitions where light line intercepts with vertical transitions for continuum onset at band edge TT

17 Exciton-Polariton photon E I II exciton k Absorption occurs at point where photon dispersion intersects exciton dispersion curve. exciton-photon interaction leads to coupled EM and polarization wave (polariton) travelling in the medium altered dispersion curve (2 branches) But: if exciton damping (phonon scattering...) is larger than excitonphoton interaction we can treat photons and excitons separately. Examples for weakly bound excitons: GaAs sub-gap excitonic absorption features exciton dissociation through collisions with LO phonons becomes more likely at higher T exciton lifetime shortened and transition line broadened Coulomb interactions increase the absorption both above and below the gap TT

18 Examples for weakly bound excitons: GaAs At low temperature (here: 1.2K) and in ultra pure material, the small line width allows observation of higher excitonic transitions: n=1 n=2 n=3 here: E g Tightly bound (Frenkel) excitons radius of weakly-bound excitons: model of bound e-h pair in dielectric medium breaks down when a n is of the order of interatomic distances (Å) have tightly bound excitons for small ε r, large µ tightly-bound electron-hole pair, typically located on same unit (atom or molecule) of the crystal (but the whole exciton may transfer through the crystal) large binding energies (0.1 1eV) excitons persist at room temperature. TT

19 Transition energies for tightly bound excitons transition energies often correspond to those found in the isolated atom or molecule that the crystal is composed of theoretical calculations may be based e.g. on tightbinding or quantum-chemical methods often need to include effects of strong coupling between excitons and the crystal lattice (polaronic contributions) Examples for tightly bound excitons: rare gas crystals absorption spectrum of crystalline Kr at 20K: E 1 =10.2eV E g =11.7eV E b =1.5eV Note: the lowest strong absorption in isolated Kr is at 9.99eV close to lowest excitonic transition E 1 in crystal TT

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21 VI Low-dimensional systems de Broglie wavelength for an electron at room temperature: growth direction z If we can make structured semiconductors on these length scales we may be able to observe quantum effects! Possible using e.g. molecular beam epitaxy (MBE) or metal-organic chemical vapour deposition (MOCVD) Al x Ga 1-x As (barrier) GaAs (well) Al x Ga 1-x As (barrier) GaAs substrate d confined holes E(k=0) valence band E g (AlGaAs) E g (GaAs) E g (AlGaAs) conduction band confined electrons Effect of confinement on the DOS Confinement in a particular direction results in discrete energy states, but free movement in other directions gives rise to continuum. Joint density of states g( ω) (for direct CB-VB transitions): 3D (bulk) 2D (Q well) 1D (Q wire) 0D (Q dot) TT

22 Quantum well with infinite potential barriers Schrödinger s eqn inside the well: outside the well: wavefunction along z: with wavevector confinement energy: Bandstructure modifications from confinement bulk GaAs (Γ-valley) E electrons J=1/2 GaAs QW E E 2,e E 1,e k Jz =3/2 heavy holes J z =1/2 light holes holes J=3/2 E 1,lh E 1,hh bandstructure for 2D motion is altered (quantum confinement leads to valence band mixing) TT

23 Optical transitions in a quantum well as before, matrix element: wavefunctions now: valence/conduction band Bloch function hole/electron wavefunction along z (i) changes slowly over a unit cell (compared to u c, u v ) (ii) unless (k-conservation) dipole transition criteria (as before) electron-hole spatial overlap in well Selection rules for optical transitions in a QW (i) wavevector conservation: (ii) parity selection rule: and must differ in parity (iii) as before need sufficient spatial overlap between electron and hole wavefunctions along the z-direction. For an infinite quantum well: unless N.B.: expect some deviation in finite quantum wells! (iv) unless and have equal parity TT

24 (v) energy conservation: hh1 e1 lh1 e1 hh2 e2 e2 e1 E g hh1 hh2 lh1 photon energy and for band gap of well material confinement energy of electron/hole kinetic energy in plane of QW (at the band edge): allowed dipole transitions Example: absorption of a GaAs/AlAs QW Absorption of GaAs/AlAs MQW (d=76å) at 4K: hh1-e1 (X) lh1-e1 (X) e1-hh3 (X) hh2-e2 (X) lh2-e2 (X) hh3-e3 (X) e3-hh1 (X) hh2-e2 (continuum) lh1-e1 (continuum) lh2-e2 (continuum) hh1-e1 (continuum) below each onset of absorption: excitonic features (X) above onset: flat absorption, since 2D joint density of states independent of ω deviation from Δn=0, in particular at high E TT

25 Influence of confinement on the exciton bulk GaAs hh lh n=1 T=300K n=2 GaAs MQW Confinement brings electron and hole closer together. enhanced exciton binding energy increased oscillator strength TT

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27 V Optical response of a free electron gas Classic Lorentz dipole oscillator model (again): solutions are as before, but with (no retaining force!) dielectric constant: with plasma frequency and background dielectric constant real and imaginary part of the dielectric constant: AC conductivity of a free electron gas Can re-write equation of motion as: electron with momentum p is accelerated by field but looses momentum at rate obtain electron velocity: and using AC conductivity where (DC conductivity) and optical measurements of ε r equivalent to those of AC conductivity! TT

28 Low-frequency regime At low frequency of the EM wave, or : and one may approximate: Skin depth (distance from surface at which incident power has fallen to 1/e): For Cu at 300K: σ 0 = Ω -1 m -1 δ = ν = 50Hz δ = ν = 100MHz High-frequency regime In a typical metal: N m -3, σ Ω -1 m -1 Drude model predicts: γ s -1 At optical frequencies: (weak damping) and (i) largely imaginary (ii) largely real wave mostly reflected wave partly transmitted, weak absorption ( ) TT

29 Reflectivity in the high-frequency regime doped semiconductors: large background dielectic constant ( ) from higher-energy interband transitions most metals: (if no strong optical transitions at higher photon energy) Example: Reflection from Alkali metals Metal N (10 28 m -3 ) ω p /2π (10 15 Hz) λ p (nm) λ UV (nm) Li Na K Rb Cs measured at low T calculated from measured UV transmission cut-off high reflectivity up to UV wavelengths good agreement between measurement and Drude-Lorentz model TT

30 Example: Reflection from transition metals Metal N (10 28 m -3 ) ω p /2π (10 15 Hz) λ p (nm) Cu Ag Au measured at low T calculated from These transition metals should be fully reflective up to deep UV But we know: Gold appears yellow, Copper red Reflection of light from Au, Cu and Al: D-L model for ω p =9eV (Au) Au Cu D-L model for ω p =15.8eV (Al) Al Drude-Lorentz model does not account fully for optical absorption of transition metals (especially in the visible) need to consider bandstructure (damping has weak effect at these frequencies) TT

31 Example: Reflection from Copper Electronic configuration of Cu: [Ar] 3d 10 4s 1 Transitions (in visible range of spectrum) between relatively dispersionless bands of tightly bound 3d electrons and half-filled band of 4s-electrons: E f 3d bands strong interband absorption for copper appears red! Example: Reflection from doped semiconductors Free-carrier reflectivity of InSb: for where Can determine effective mass of majority carriers from free carrier absorption TT

32 Example: Free-carrier absorption in semiconductors For free carriers in the weak absorption regime ( ): predict: But experiments on n-type samples show: where Deviations arise from: intraband transitions involving phonon scattering in p-type semiconductors: intervalence band absorption absorption by donors bound to shallow donors or acceptors Example: Impurity absorption in semiconductors In doped semiconductors the electron (hole) and the ionized impurity are attracted by Coulomb interaction hydrogenic system Absorption of Phospor-doped silicon at 4.2K: conduction band n=2 n= n=1 valence band Observe Lyman series for transitions from 1s level of Phosphor to p levels, whose degeneracy is lifted as a result of the anisotropic effective mass of the CB in Si TT

33 Plasmons At the plasma edge ( ): What happens at this frequency? Polarization induced by the EM wave: where Wavevector P is equal and opposite to incident field At the Plasma edge a uniform E-field in the material shifts the collective electron w.r.t the ionic lattice! Plasma oscillations for ε r =0: applied EM wave resulting charge distribution induced macroscopic polarization For a (transverse) wavevector, the resulting charge distribution corresponds to a longitudinal oscillation of the electron gas with frequency ω p! The quantum of such collective longitudinal plasma oscillations is termed a plasmon. TT

34 Example: Plasmons in n-type GaAs Light scattered from n-type GaAs at 300K: plasmon emission plasmon absorption Energy conservation: from data: Expect from: TT

35 VI Optical studies of phonons Dispersion relation for a diatomic linear chain: transverse optical (TO) phonon: 0 light dispersion optical branches (1LO + 2TO) acoustic branches (1LA + 2TA) transverse acoustic (TA) phonon: EM radiation is a transverse wave with wavevector and can thus interact directly only with TO modes in polar crystals near the centre of the Brillouin zone. Harmonic oscillator model for the ionic crystal lattice Diatomic linear chain under the influence of an external electric field: Equations of motion: x - x + where frequency of TO mode near centre of Brillouin zone (with effective spring constant C) reduced mass relative displacement of positive and negative ions TT

36 Add damping term to account for finite phonon lifetime: Displacement of ions induces polarization P = NQx Dielectric constant (as before): Rewrite this result in terms of the static ( ) and the high-frequency ( ) limits of the dielectric constant: Lattice response in the low-damping limit Long phonon lifetimes: Consider Gauss s law. In the absence of free charge: or wave must be transverse ( ) longitudinal wave possible ( ) TT

37 What happens at ε r =0? Again: Wavevector of EM wave in medium: all ions of same charge shift by the same amount throughout the medium result can be seen as a transverse wave ( ) with k 0 or as a longitudinal wave ( ) in orthogonal direction. The Lyddane-Sachs-Teller relationship At ε r =0 the induced polarization corresponds to a longitudinal wave, i.e. ε r (ω LO )=0 Lyddane-Sachs-Teller relationship And from follows: In polar crystals the LO phonon frequency is always higher than the TO phonon frequency In non-polar crystals, and the LO and TO phonon modes are degenerate (at the Brillouin zone centre) TT

38 Dielectric constant and Reflectivity for undamped lattice Reststrahlen band (R=1) Influence of damping Lattice Reflectivity: For finite phonon lifetime (γ 0) at resonance: Reststrahlen band no longer fully reflective general broadening of features TT

39 Measurements of IR reflectivity Fourier transform infrared spectroscopy (FTIR): measure interference pattern I(d) as a function of mirror displacement d I(d) gives Fourier transform of sample transmission T(ν) multiplied with system response S(ν): detector sample IR source M2 d M1 Example: reflection spectra for zinc-blende-type lattices measured at 4.2K measured at RT TT

40 Phonon-Polaritons Examine more closely the dispersion relations for phonons and the EM wave near the Brillouin zone centre: If no coupling occurred: But: coupling between TO phonon and EM wave leads to modified dispersion resulting wave is mixed mode with characteristics of TO polarization and EM wave LO phonon dispersion remains unchanged as it does not couple to the EM wave Phonon-Polariton dispersion: upper polariton branch two branches for ω(k) Reststrahlen band (perfect reflection, no mode can propagate) lower polariton branch TT

41 Inelastic Light Scattering Scattering of light may be caused by fluctuations of the dielectric susceptibility χ of a medium time-dependent variation of χ may be caused by elementary excitations, e.g. phonons or plasmons scattering from optical phonons is called Raman scattering and that from acoustic phonons Brillouin scattering if u(r,t) is the displacement (of charge) associated with the excitation, the susceptibility can be expressed in terms of a Taylor series: Polarization in the medium: let light wave with frequency ω lattice wave with frequency ω q where unscattered polarization wave Anti-Stokes scattering Stokes scattering TT

42 Energy & momentum conservation for inelastic scattering Scattering process: ω in, k ω out, k out in ω q, q energy conservation: wave vector conservation: Anti-Stokes scattering requires absorption of a phonon and therefore sufficiently high temperature. In general the ratio of Anti-Stokes to Stokes scattering intensities is given by: Maximum momentum transfer in backscattering geometry, where: Inelastic light scattering probes phonons with small wave vector Raman spectroscopy: Experimental details Require detection of optical phonons within typical frequency range 1cm -1 < ω p < 3000cm -1 need excitation source (laser) with sufficiently narrow bandwidth need detection system with high dispersion and ability to suppress elastically scattered light Typical set-up: filter polarizer sample Laser detector G2 M3 analyzer G1 S4 S1 S2,3 M5 M4 M2 M1 doublegrating spectrometer TT

43 Raman spectra for zinc-blende-type semiconductors Brillouin scattering: Experimental details Require detection of acoustic phonons near the centre of the Brillouin zone where ω q =v ac q need to be able to measure shifts of only a few cm -1! Set-up based on a Multipass Interferometer: Brillouin spectrum for Si(100): TT

44 Phonon lifetimes Experimental evidence for finite phonon lifetimes from i. Reflectivity measurements: R<1 in Reststrahlen band ii. Raman scattering: non-zero width of Raman line Data suggests phonon lifetimes of 1-10ps in typical inorganic semiconductors. Origin of short phonon lifetimes: anharmonic potential experienced by the atoms: Anharmonic terms make possible higher-order processes, e.g. phonon-phonon scattering: optical branch acoustic branch TT

45 VII Optics of anisotropic media A medium is anisotropic if its macroscopic optical properties depend on direction Examples: isotropic anisotropic gas, liquid, amorphous solid polycrystalline crystalline liquid crystal ε r and χ in an anisotropic medium Polarizability now depends on direction in which E-field is applied relative electric permittivity ε r and susceptibility χ now tensors: or D and E no longer necessarily point into the same direction! But can always find coordinate system for which off-diagonal elements vanish, in which case: In the directions of these principal crystal axes E and D are parallel. TT

46 Propagation of plane waves in an isotropic medium Ampere s and Faraday s law for plane waves: where is the wavevector in free space. Choosing a coordinate system along the crystal s principal axes yields: homogeneous matrix equation, require Solving the matrix equation ( ) yields: This provides a dispersion relationship Can obtain the refractive index from the ratio of phase velocities in vacuo and inside medium: TT

47 Propagation of plane waves in uniaxial crystals In uniaxial crystals (optic axis along z):, Two solutions: (i) Sphere: for ordinary ray (polarized to k-z plane) (ii) ellipsoid of revolution k 3 k-direction n o k 0 n(θ)k n o k 0 0 θ n o k 0 n e k 0 k2 for extraordinary ray (polarized in k-z plane) Ordinary vs extraordinary rays in uniaxial crystals n o k 0 k 3 S n o k 0 k 3 S H E,D θ k n o k 0 k 2 E D θ H k n e k 0 k 2 (a) ordinary ray: E, D polarized to plane containing k and the optic axis; Refractive index: (b) extraordinary ray: E, D polarized in plane containing k and the optic axis TT

48 Comments on wave propagation in uniaxial crystals 1) Faraday s & Ampere s law for plane waves in dielectrics: D is normal to both k and H H is normal to both k and E N.B.: this does not imply E k! 2) All fields are of the form wavefronts are to k. 3) The phase velocity v is in the direction of k with 4) As usual, the group velocity is v g is normal to the k-surface! 5) The pointing vector is normal to E and H Can show: for small Δk S normal to k-surface 6) From (5) and (1) follows that E is parallel to the k-surface. k 0 e-ray ϕ α Refraction at the surface of a uniaxial crystal β k crystal air optic axis phase matching condition: N.B.: Snell s law holds for the directions of k in the media, but this is not necessarily the direction of ray propagation! Example: Double refraction at normal incidence: TT

49 Linear optics: VIII Non-linear Optics Polarization depends linearly on the electric field: Electrons experience harmonic retaining potential But: refractive index n, absorption coefficient α, reflectivity R independent of incident EM wave s intensity If E-fields become comparable to those binding electrons in the atom, anharmonic (non-linear) effects become significant. For an H-atom: need EM wave intensity Possible with tightly focused laser beams! The non-linear susceptibility tensor For a medium in which we may in general write: linear non-linear part now power-dependent! In an anisotropic medium, non-linear response will depend on directions of E-fields wrt the crystal: Second-order non-linear polarization components: Third-order non-linear polarization components: TT

50 Non-linear medium response to sinusoidal driving field low applied field amplitude: high applied field amplitude: If and the applied field, then: second-order nonlinearity: rectification and frequency doubling third-order nonlinearity: frequency tripling Second-order nonlinearities (NL) Treatment of non-resonant 2 nd order NL within oscillator model: Assume anharmonic potential: Equation of motion: Use trial solution: Assume Obtain displacement amplitudes: TT

51 Calculating the induced polarization: linear susceptibility, as before second-order non-linear susceptibility Can re-write the second-order non-linear susceptibility as: materials with large linear susceptibilty also have a large non-linear susceptibility in a centrosymmetric medium, U(x) = U(-x) and therefore C 3 = 0 and χ (2) =0 second-order nonlinearities only occur in media that lack inversion symmetry! (This may also be shown directly from the definition of P(2) - see question sheet.) TT

52 The second-order non-linear coefficient tensor d ij 27 components in χ (2) ijk But some of these components must be the same (e.g. χ (2) xyz E y E z = χ(2) xzy E z E y, so χ(2) xyz = χ(2) xzy because ordering of fields is arbitrary) Second-order response can be described by the simpler tensor d ij, i.e. In many cases crystal symmetry requires that most of the components of d ij vanish. 2 nd order NL: Frequency (three-wave) mixing Presume two waves are travelling in the medium, with The induced polarization is: sum-frequency generation difference-frequency generation Feynman diagrams for second-order nonlinear frequency mixing: ω 1 ω 1 + ω 2 ω 1 ω 1 - ω 2 ω 2 - ω 2 TT

53 2 nd order NL: Frequency doubling Consider the generation of second harmonics in more detail: Maxwell s equations: Wave equation: Consider propagation of second-harmonic wave in z-direction: Let this wave be generated from two fundamental waves: Obtain specific wave equation: Assume that the variation of the complex field amplitude is small (slowly varying envelope approximation): Obtain DE for increase of the second harmonic along the direction of propagation: where is the phase mismatch between fundamental and second harmonic wave TT

54 2 nd order NL: Phase matching conditions E(ω) E(ω) E(2ω) 0 L For efficient frequency conversion, we need the fundamental wave and the higher harmonic to be in phase throughout the crystal, i.e. where The second-harmonic field at length L for arbitrary is: For constant fundamental wave amplitudes (thin crystal) the second harmonic intensity is then given by: Second-harmonic intensity after propagation through crystal of length L without phase matching First intensity minimum at: But: dispersion in media means that in general: Example: Sapphire need too thin a crystal to achieve efficient 2 nd harmonics generation TT

55 2 nd order NL: Phase matching in a uniaxial crystal In general, in the birefringent medium, but since the refractive index now depends on the direction of propagation and wave polarization wrt the optic axis, for some geometries we may have phase matching! n o 2ω n o ω θ phase-matching direction n e ω n e 2ω Here, phase matching occurs for the fundamental travelling as extraordinary (polarization in plane) and the 2 nd harmonic as ordinary (polarization to plane) with Third-order nonlinearities Third-order effects become important in centrosymmetric (e.g. isotropic media) where For three waves with frequencies nonlinear polarization is the third-order generates a wave with (a) four-wave mixing ω2 ω 1 ω 3 ω ω -ω ω 1 + ω 2 + ω 3 (c) Optical Kerr effect ω (b) frequency tripling ω ω -ω ω S ω S 3ω e.g.: (d) stimulated Raman scattering TT

56 3 rd order NL: The optical Kerr effect Optical Kerr effect: and no phase mismatch In an isotropic medium (χ (2) =0): Refractive index: where light intensity Refractive index varies with light intensity! Example: Kerr lensing Optical Kerr effect: at high intensity I Laser beam with spatially varying profile (e.g. Gaussian beam) experiences in the medium a higher refractive index at the centre of the beam than the outside medium acts as a lens! n(x) x Propagation of an intense gaussian beam through a Kerr medium: TT

57 3 rd order NL: Resonant nonlinearities hν Consider medium with optical transition at resonance with incident wave of Intensity I find that absorption decreases as higher state become more populated: Absorption coefficient for for (weak absorption) Can view saturable absorption as a third-order optical nonlinearity! TT